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4. 5. 6. 1# = 1, (ta)# = t2 a# , N(ta) = t3 N(a), T(a, a# ) = 3N(a), a## = N(a)a, b = T(b, 1) · 1 − 1 × b, 7. 8. 9. 10. 11. 12. T(a × b, c) = T(a, b × c), N(a + b) = N(a) + T(a# , b) + T(a, b# ) + N(b), (a + b)# = a# + a × b + b# , a# × (a × b) = N(a)b + T(a# , b)a, a# × b# + (a × b)# = T(a# , b)b + T(a, b# )a, N(a) = 0 if and only if a = 0. If we define the inverse a−1 of an arbitrary nonzero a ∈ J as a−1 = N(a)−1 a# , then we can define the Moufang set MH(J) related to J in exactly the same way as before for the projective line over a field K, in its non-homogeneous representation.

No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Table 1: M22 from Method 1 Relators Length Total cosets aababAAB, abbbbaBaB 17 21611026 aaaaabbb, aababABABab 19 23024264 aaaaa, bbb, aababABABab 19 12902711 aaaaabbb, aabABababAB 19 24442031 aaaaa, bbb, aabABababAB 19 13063356 aababAAB, aaaaaabbbbb 19 40304685 aababAAB, aaaaaa, bbbbb 19 17917189 aaaabAbAb, aabABabbAB 19 23098382 aababABAB, abbabbaBBB 19 28017778 aaaaa, ababab, abbAbABB 19 11181678 abc, aaBcAb, acccBCaC 19 19102618 abc, aaBcbb, acBcBCCC 19 19426579 aabAABB, aaabbabAbAbAb 20 29179041 aabAABB, aabaBABABABab 20 22226752 aabAABB, ababAbbABBBAb 20 20068916 aabaabAAB, ababababaBB 20 24018995 aaaaa, ababab, aabABBabAB 21 13063072 aaaaa, ababab, abaBaBaBBB 21 38353459 aaaaa, ababab, abbAbAbbbb 21 37692724 The presentations in Table 1 should be considered in the context of the following three results about relator amalgamation which appear in [4] with proofs and various applications.

Debroey, Semi partial geometries satisfying the diagonal axiom, J. Geom. 13 (1979), 171–190. [5] I. Debroey and J. A. Thas, On semipartial geometries, J. Combin. Theory Ser. A 25 (1978), no. 3, 242–250. [6] F. De Clerck and H. Van Maldeghem, Some classes of rank 2 geometries, in: Handbook of Incidence Geometry, Buildings and Foundations (ed. F. Buekenhout), Chapter 10, North-Holland (1995), 433–475. [7] C. Hering, W. M. Kantor and G. M. Seitz, Finite groups with a split BN-pair of rank 1, I, J.

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A bisection algorithm for the numerical Mountain Pass by Barutello V., Terracini S.


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